
theorem Th7:
for F be FinSequence of COMPLEX, Fr be FinSequence of REAL st Fr=Re F
  holds Sum(Fr) = Re Sum(F)
proof
defpred P[Nat] means
  for F be FinSequence of COMPLEX,
      Fr be FinSequence of REAL
      st len F = $1 & Fr=Re F
    holds Sum(Fr) = Re Sum(F);
A1: P[0]
  proof
    let F be FinSequence of COMPLEX,
        Fr be FinSequence of REAL;
    assume A2: len F = 0 & Fr=Re F;
    then dom Fr = dom F by COMSEQ_3:def 3
          .= Seg len F by FINSEQ_1:def 3; then
  A3: len Fr = 0 by A2,FINSEQ_1:def 3;
    thus Re (Sum (F)) = Re (0) by A2,Lm2,FINSEQ_1:20
                     .= Sum Fr by A3,COMPLEX1:4,FINSEQ_1:20,RVSUM_1:72;
  end;
A4:now let k be Nat;
     assume A5:P[k];
     now let F be FinSequence of COMPLEX,
             Fr be FinSequence of REAL;
       assume A6:len F = k+1 & Fr=Re F;
       reconsider F1= F|k as FinSequence of COMPLEX;
       A7: len F1 = k by A6,FINSEQ_1:59,NAT_1:11;
       reconsider F1r= Re F1 as FinSequence of REAL;
       reconsider x=F.(k+1) as Element of COMPLEX by XCMPLX_0:def 2;
       A8: F = F1^ <* x *> by A6,FINSEQ_3:55;
       hence Re (Sum(F)) = Re (Sum(F1)+ x ) by Lm3
          .=Re (Sum(F1)) + Re x by COMPLEX1:8
          .=Sum (F1r) + Re x by A5,A7
          .=Sum(F1r^<* Re x *> ) by RVSUM_1:74
          .=Sum(Fr) by A6,A8,Th5;
     end;
     hence P[k+1];
   end;
A9:for k be Nat holds P[k] from NAT_1:sch 2(A1,A4);
let F be FinSequence of COMPLEX,
    Fr be FinSequence of REAL;
assume A10: Fr=Re F;
len F is Element of NAT;
hence Sum(Fr) = Re Sum(F) by A9,A10;
end;
